Uniqueness of meromorphic functions that share three values
The author proves the following theorem on shared values of meromorphic functions. Theorem. Let \(f\) and \(g\) be two distinct nonconstant meromorphic functions such that \(f\) and \(g\) share 0, 1, \(\infty\) with the same multiplicities (i.e. C.M.), and let \(a_ 1,a_ 2,\dots,a_ p\) be \(p\) (\(\geq 1\)) distinct finite complex numbers, and \(a_ i\neq 0\) (\(i=1,2,\dots,p\)). If \[ \sum_{i=1}^ p f(a_ i,f)+f(\infty,f)>{{2(p+1)} \over {p+2}}, \] then there exists one and only one \(a_ j\) in \(a_ 1,a_ 2,\dots,a_ p\) such that \(a_ j\) and \(1- a_ j\) are Picard exceptional values of \(f\) and \(g\) respectively, and also \(\infty\) is so, and \[ (f-a_ j)(g+a_ j-1)\equiv a_ j(1-a_ j). \] {}.
scientific article; zbMATH DE number 167942